| Exam Board | CAIE |
|---|---|
| Module | P3 (Pure Mathematics 3) |
| Year | 2006 |
| Session | November |
| Marks | 9 |
| Paper | Download PDF ↗ |
| Mark scheme | Download PDF ↗ |
| Topic | Implicit equations and differentiation |
| Type | Find stationary points |
| Difficulty | Standard +0.3 This is a straightforward implicit differentiation question with standard techniques. Part (i) requires routine application of implicit differentiation rules, and part (ii) involves setting dy/dx = 0 and solving the resulting system—both are textbook exercises with no novel insight required. Slightly above average difficulty due to the algebraic manipulation needed, but well within standard P3 expectations. |
| Spec | 1.07s Parametric and implicit differentiation |
| Answer | Marks |
|---|---|
| (i) State \(2(3y^2)\frac{dy}{dx}\) as derivative of \(2y^3\), or equivalent | B1 |
| State \(3x \frac{dy}{dx} + 3y\) as derivative of \(3xy\), or equivalent | B1 |
| Solve for \(\frac{dy}{dx}\) | M1 |
| Obtain given answer correctly | A1 |
| [The M1 is dependent on at least one of the B marks being obtained.] | |
| (ii) State or imply that the coordinates satisfy \(y - x^2 = 0\) | B1 |
| Obtain an equation in \(x\) (or in \(y\)) | M1 |
| Solve and obtain \(x = 1\) only (or \(y = 1\) only) | A1 |
| Substitute \(x\)- (or \(y\)-)value in \(y - x^2 = 0\) or in the equation of the curve | M1 |
| Obtain \(y = 1\) only (or \(x = 1\) only) | A1 |
| [SR: If B1 is earned and (1, 1) stated to be the only solution with no other evidence, award B2. If the point is also shown to lie on the curve award a further B2.] | |
| 5 |
(i) State $2(3y^2)\frac{dy}{dx}$ as derivative of $2y^3$, or equivalent | B1 |
State $3x \frac{dy}{dx} + 3y$ as derivative of $3xy$, or equivalent | B1 |
Solve for $\frac{dy}{dx}$ | M1 |
Obtain given answer correctly | A1 |
[The M1 is dependent on at least one of the B marks being obtained.] | | |
(ii) State or imply that the coordinates satisfy $y - x^2 = 0$ | B1 |
Obtain an equation in $x$ (or in $y$) | M1 |
Solve and obtain $x = 1$ only (or $y = 1$ only) | A1 |
Substitute $x$- (or $y$-)value in $y - x^2 = 0$ or in the equation of the curve | M1 |
Obtain $y = 1$ only (or $x = 1$ only) | A1 |
[SR: If B1 is earned and (1, 1) stated to be the only solution with no other evidence, award B2. If the point is also shown to lie on the curve award a further B2.] | | |
| | 5 |6 The equation of a curve is $x ^ { 3 } + 2 y ^ { 3 } = 3 x y$.\\
(i) Show that $\frac { \mathrm { d } y } { \mathrm {~d} x } = \frac { y - x ^ { 2 } } { 2 y ^ { 2 } - x }$.\\
(ii) Find the coordinates of the point, other than the origin, where the curve has a tangent which is parallel to the $x$-axis.
\hfill \mbox{\textit{CAIE P3 2006 Q6 [9]}}